Using a result linking convexity and irreducibility of matrix sets it is shown that the generalized spectral radius of a compact set of matrices is a strictly increasing function of the set in a very natural sense. As an application some consequences of this property in the area of time-varying stability radii are discussed. In particular, using the implicit function theorem sufficient conditio...

A b s t r a c t. Let K3 and K ′ 3 be two complete graphs of order 3 with disjoint vertex sets. Let B∗ n(0) be the 5-vertex graph, obtained by identifying a vertex of K3 with a vertex of K ′ 3 . Let B∗∗ n (0) be the 4-vertex graph, obtained by identifying two vertices of K3 each with a vertex of K ′ 3 . Let B∗ n(k) be graph of order n , obtained by attaching k paths of almost equal length to the...

A decomposition result for planar graphs is used to prove that the spectral radius of a planar graph on n vertices is less than 4 + 3(n 3) Moreover, the spectral radius of an outerplanar graph on n vertices is less than 1 + JZ+&-X

Let G be a graph with n vertices, m edges, girth g, and spectral radius μ. Then

If ρ(A) > 1, then lim n→∞ ‖A‖ =∞. Proof. Recall that A = CJC−1 for a matrix J in Jordan normal form and regular C, and that A = CJnC−1. If ρ(A) = ρ(J) < 1, then J converges to the 0 matrix, and thus A converges to the zero matrix as well. If ρ(A) > 1, then J has a diagonal entry (J)ii = λ n for an eigenvalue λ such that |λ| > 1, and if v is the i-th column of C and v′ the i-th row of C−1, then ...

The spectral radius of a matrix A is the maximum norm of all eigenvalues of A. In previous work we already formalized that for a complex matrix A, the values in A grow polynomially in n if and only if the spectral radius is at most one. One problem with the above characterization is the determination of all complex eigenvalues. In case A contains only non-negative real values, a simplification ...